Divided Differences and Polynomial Convergences

The continuous analysis, such as smoothness and uniform convergence, for polynomials and polynomial-like functions using differential operators have been studied considerably, parallel to the study of discrete analysis for these functions, using difference operators. In this work, for the difference operator <TEX>∇ₕ</TEX> with size h > 0, we verify that for an integer <TEX>m≥0</TEX> and a strictly decreasing sequence <TEX>hₙ</TEX> converging to zero, a continuous function f(x) satisfying <TEX>∇hₙᵐ⁺¹f(khₙ)=0, for every n≥1 and k∈Z</TEX>, turns to be a polynomial of degree <TEX>≤m</TEX>. The proof used the polynomial convergence, and additionally, we investigated several conditions on convergence to polynomials.